Checking DL-Lite modularity with QBF solvers

Checking DL-Lite modularity with QBF solvers

R. Kontchakov¹, V. Ryzhikov², F. Wolter³, and M. Zakharyaschev¹

1 School of Computer Science and Information Systems, Birkbeck College, London {roman,michael}@dcs.bbk.ac.uk

2 Faculty of Computer Science, Free University of Bozen-Bolzano, Italy ryzhikov@inf.unibz.it

3 Department of Computer Science, University of Liverpool, U.K., frank@csc.liv.ac.uk

Abstract. We show how the reasoning tasks of checking various versions of conservativity for the description logicDL-Litebool can be reduced to satisfiability of quantified Boolean formulas and howoff-the-shelf QBF solvers perform on a number of typicalDL-Litebool ontologies.

1 Introduction

Recently, the notion of conservative extension and variants thereof have been identified as fundamental for developing, re-using and maintaining ontologies [1–4]. Intuitively, an ontology T₁₂ is a conservative extension of an ontologyT₁ w.r.t. a signatureΣifT₁₂⊇ T₁and bothT₁₂andT₁ provide precisely the same information about Σ in the sense that every ‘Σ-formula’ derivable from T₁₂ is derivable fromT1 as well. If this happens to be the case, then

– an ontology engineer interested only inΣ can useT1 instead of the possibly much largerT12, and

– an ontology engineer who has addedT12\ T1 to T1 can be certain that this addition does not change (or ‘damage’) the meaning assigned toΣ byT1. For further discussion and applications the reader is referred to [3, 5].

In the ‘definition’ of conservativity above, we did not specify the language of Σ-formulas. In fact, different applications may require different languages:

one might be only interested in implications between concept names (concept classification), implications between complex concepts, answers to queries when the ontology is used to query databases, etc., which leads to different notions of conservativity. In [6], several such notions were introduced and investigated for the DL-Lite family of description logics. It was shown, in particular, that the complexity of checking conservativity sits betweencoNPandΠ₂^p, depending on the member of the DL-Lite family and the type of Σ-formulas. Note that for propositional logic deciding conservativity corresponds to deciding validity of quantified Boolean formulas (QBFs) of the form ∀p∃qϕ, and the lower bounds established in [6] follow from the corresponding lower bounds for QBFs.

The purpose of this paper is to report on (i) how the semantic conservativity criteria found in [6] can be refined in such a way that one can usegeneral purpose

off-the-shelf QBF solvers for deciding conservativity, and (ii) how different QBF solvers perform on a number of ‘typical’DL-Liteontologies.

The paper is organised in the following way. In the next section we remind the reader of the logicDL-Litebool, briefly discuss the conservativity notions from [6] and illustrate them with an illuminative example. In Section 3 we provide a semantic criterion of deductive conservativity, show how it can be encoded by means of QBFs, and report on the results of our experiments with three standard QBF solvers: sKizzo [7], 2clsQ [8] and Quaffle [9, 10]. Section 4 deals with query conservativity, and in Section 5 we discuss the obtained results and future work.

2 Conservativity in DL-Lite

Our main concern in this paper is the logicDL-Litebool [11] which covers most of the members of theDL-Litefamily [12, 13, 11]. The language ofDL-Litebool has object names a1, a2, . . ., concept names A1, A2, . . ., androle names P1, P2, . . .. Complexroles Rand concepts Care defined as follows:

R ::= P_i | P_i⁻, B ::= ⊥ | A_i | ≥q R, C ::= B | ¬C | C1uC2, whereq≥1. (Other usual concept constructs like>,∃R,≤q RandC1tC2can be used as standard abbreviations.) Aconcept inclusion is of the formC1vC2, where C1 and C2 are concepts; a DL-Litebool TBox is a finite set of concept inclusions, and anABox is a set of assertions of the formC(ai),R(ai, aj), where C is a concept,R a role, anda_i, a_j are object names. Aknowledge base (KB) is a pair (T,A) consisting of a TBoxT and an ABoxA.

The semantics of DL-Lite_bool is defined in the usual way using interpreta- tionsI= (∆^I, A^I₁, . . . , P₁^I, . . . a^I₁, . . .). An (essentially positive)existential query q(x₁, . . . , x_n) is a first-order formula of the form

∃y1. . .∃ymϕ(x1, . . . , xn, y1, . . . , ym),

where ϕis constructed, using only∧ and ∨, from atoms of the formC(s) and P(s1, s2), withCbeing a concept,P a role, andsieither a variable from the list x1, . . . , xn, y1, . . . , ym or an object name. Given a KB (T,A) and a queryq(x), with x = x1, . . . , xn, we say that an n-tuple a of object names is an answer to q(x) w.r.t. (T,A) and write (T,A)|=q(a) if, for every model I for (T,A), we have I |= q(a). The data complexity of the query answering problem for DL-Lite_bool KBs iscoNP-complete [11].

A signature Σ is a finite set of concept and role names. Given a concept (role, TBox, ABox, query)E, we denote by sig(E) thesignature ofE, i.e., the set of concept and role names that occur inE. Note that⊥and>are regarded aslogical symbols. A concept (role, TBox, ABox, query)Eis a Σ-concept (role, TBox,ABox,query, respectively) ifsig(E)⊆Σ. Thus,P⁻ is aΣ-role iffP ∈Σ.

We now introduce four types of conservativity we deal with in this paper; for further details and discussion we refer the reader to [6].

Definition 1. LetT₁⊆ T₁₂ beDL-Lite_bool TBoxes andΣ a signature.

– T12 is called a deductive conservative extension of T1 w.r.t. Σ if, for every concept inclusionC1 vC2 withsig(C1 vC2)⊆Σ, we have T1 |=C1vC2

wheneverT12|=C₁vC₂.

– T12 is a query conservative extension of T1 w.r.t. Σ if, for every ABox A withsig(A)⊆Σ, every queryqwithsig(q)⊆Σ, and every tupleaof object names fromA, we have (T1,A)|=q(a) whenever (T12,A)|=q(a).

– T12 is astrong deductive (query)conservative extension ofT1in w.r.t.Σ if T12∪ T is a deductive (respectively, query) conservative extension ofT1∪ T w.r.t.Σ, for every TBoxT withsig(T)∩sig(T₁₂)⊆Σ.

Theorem 1 ([6]). For any twoDL-Litebool TBoxesT1⊆ T12 and signatureΣ, T12 is a query conservative extension ofT1 w.r.t.Σ iffT12 is a strong deductive conservative extension ofT1w.r.t.ΣiffT12is a strong query conservative exten- sion ofT1w.r.t.Σ. Every query conservative extension is a deductive conservat- ive extension, but not the other way round. The problems of deciding deductive and query conservativity are bothΠ₂^p-complete.

We illustrate this theorem by an example which shows that checking conser- vativity is a non-trivial task, even for a transparent ontology with 10 axioms.

Example 1. LetΣ={teaches},T1 contain the axioms

ResearchStaffuVisiting v ⊥, Academic v ∃teachesu ¬ ≥2teaches,

∃teaches v AcademictResearchStaff, ∃writes v AcademictResearchStaff, ResearchStaff v ∃worksIn, ∃worksIn⁻ v Project,

Project v ∃manages⁻, ∃manages v AcademictVisiting, and letT12=T1∪ {Visitingv ≥2writes}.It is not hard to see that inT12we can derive new inclusions like Visitingv ∃teachesu ¬ ≥2teaches, but nothing new in the signature Σ. It follows thatT12 is a deductive conservative extension of T1 w.r.t.Σ. Consider now the ABoxA = {teaches(a, b),teaches(a, c)} and the query q = ∃x (∃teaches)(x)∧(¬ ≥2teaches)(x)

, that is, ‘is there anybody who teaches precisely one module?’ Clearly, (T1,A)6|=qbecauseI |= (T1,A) and I 6|=q, forIwith domain{a, b, c, u, v}andAcademic^I=∅,ResearchStaff^I={a}, Visiting^I ={v}, Project^I ={u}, teaches^I ={(a, b),(a, c)}, worksIn^I ={(a, u)}, manages^I={(v, u)}. On the other hand, (T12,A)|=q. Indeed, letI |= (T12,A).

Then a ∈ ResearchStaff^I, and so there is u such that (a, u) ∈ worksIn^I and u∈Project^I. Then we have somevwith (v, u)∈manages^I andv∈(Academict Visiting)^I. Clearly, T₁₂ |= Visiting v Academic, from which v ∈ Academic^I. It follows that there is w such that (v, w) ∈ teaches^I and that such a point is unique. Therefore,T₁₂ is not a query conservative extension ofT₁w.r.t.Σ.

Note that the queryqabove contains thenegatedconcept¬ ≥2teaches. This is permitted by our definition of query conservativity, although one might argue that most query languages used in DL only allow positive existential queries.

The reason we adopt a more ‘liberal’ definition is that it gives us a more robust

notion of conservativity, which isstable under the addition of ‘abbreviations’ to an ontology (e.g., the addition of A ≡ ¬ ≥2teaches to T₁ and A to Σ should not affect conservativity). For our definition of queries, this is trivially the case, but it is not so for any smaller class of queries.

By Theorem 1,T12 should not be a strong deductive conservative extension ofT1w.r.t.Σeither. Indeed, letT = {∃teachesv ≥2teaches}. ThenT1∪T2∪T is not a deductive conservative extension ofT1∪ T w.r.t.Σbecause ∃teachesis not satisfiable w.r.t.T12∪ T, but is satisfiable w.r.t.T1∪ T.

3 Deductive conservativity

Let us fix TBoxesT1⊆ T12 and a signatureΣ⊆Σ1⊆Σ12, whereΣ1=sig(T1) andΣ12=sig(T12).⁴Letm0andm_T be, respectively, the number of role names inΣ andT, forT ∈ {T1,T12}. Denote byQthe set of all numerical parameters (together with 1) that occur inT12.⁵

Without loss of generality we will assume that bothT₁ and T₁₂ contain all the axioms of the form≥q⁰R v ≥q R, for all rolesRin T andq, q⁰∈Qsuch that qis the immediate predecessor ofq⁰. We will also assume that our TBoxes contain only axioms of the form D1 v D2, where the Di are conjunctions of concepts of the formB or¬B from the definition ofDL-Litebool concepts.⁶

Let Σ₀ ∈ {Σ, Σ₁, Σ₁₂}. A Σ₀Q-concept is any concept of the form ⊥, A_i,

≥q R, or its negation, for someA_i∈Σ₀,Σ₀-roleRandq∈Q. AΣ₀Q-type is a settofΣ₀Q-concepts containing>and such that the following holds:

– for everyΣ0Q-conceptC, eitherC∈tor¬C∈t(but not both), – ifq < q⁰ are both inQand≥q⁰R∈tthen≥q R∈t,

– ifq < q⁰ are both inQand¬(≥q R)∈tthen¬(≥q⁰R)∈t.

For a type t, there is always an interpretation I and a point xin it such that x∈C^I, for allC∈t. In this case we say thattisrealised at xin I.

Definition 2. A set Ξ of Σ0Q-types is T-realisable if there is a model for T realising all types in Ξ. Ξ isprecisely T-realisable if there is a modelI for T such thatI realises all types in Ξ and everyΣ0Q-type realised inI is in Ξ.

The following semantic conservativity criterion was proved in [6]:

Theorem 2. T12 is a deductive conservative extension ofT1 w.r.t. Σ iff every T1-realisable ΣQ-type is T12-realisable.

4 AsDL-Liteboolhas the interpolation property [6], one can always assumeΣ⊆sig(T1).

5 In the QBF translations below, instead ofQwe use the setsQR of numerical para- meters, for each individual roleR.

6 Clearly, every DL-Litebool TBox can be equivalently transformed into this form.

Note, however, that the number of auxiliary variablesw^jwe will need to transform QBFs into prenex CNFs depends on this particular representation.

We now refine the criterion of Theorem 2 with the aim of encoding it by means of QBFs.T-realisability of a type tmeans that there is a preciselyT-realisable set Ξ of types at least one of which expands t. And it turns out that one can always find such aΞof size≤m_T+ 1. Moreover, we can order the types inΞin such a way that itsi’s typeti ‘takes care of the rolePi.’ To make this claim more precise we need a definition. For a ΣQ-typet, a sequenceΘ^T_t =t0,t1, . . . ,tm_T

of (not necessarily distinct)sig(T)Q-types is called aT-witness set for tif (a1) t⊆t0;

(b₁) each type int₀,t₁, . . . ,t_mT isT-realisable;

(c1) ∃Pi∈tj, for somej, iff∃Pi∈ti or∃P_i⁻∈ti, for each of the role namesPi

inT, 1≤i≤m_T.

Theorem 3. A ΣQ-type t is T-realisable iff there is aT-witness set for t. So T₁₂ is a deductive conservative extension ofT₁ w.r.t.Σ iff, for everyΣQ-typet, whenever there is aT1-witness set fortthen there is also aT12-witness set fort.

To translate the criterion of Theorem 3 into QBF, with each basic Σ0Q- concept (different from⊥) we associate a propositional variable. Fix some linear order on the set of allDL-Lite_bool basic concepts, and letB₁, . . . , B_n be the in- duced list ofΣ₀Q-concepts. Then any vectort= (b₁, . . . , b_n) of distinct propos- itional variablesb_i can be used to encodeΣ₀Q-types: every classical assignment a(of the truth valuesFandTto propositional variables) gives rise to theΣ₀Q- typet^a(t) such thatBi ∈t^a(t) iffa(bi) =T(and so ifa(bi) =Fthen¬B∈t^a(t)).

We will call t a Σ0Q-vector and t^a(t) the Σ0Q-type of t under a. We also set t(Bi) =bi and extend this map inductively to complexΣ0Q-concepts:

t(⊥) =⊥, t(¬C) =¬t(C), t(C1uC2) =t(C1)∧t(C2).

We use concatenationt0·t1of typest0,t1(when extendingΣ0Q-types toΣ₀⁰Q- types, Σ0 ⊂ Σ₀⁰) and projection t{B1,...,Bk} = (t(B1), . . . ,t(Bk)) (not a Σ0Q- vector, in general). A sequencetⁿ, . . . ,t^mofΣQ0-vectors is denoted byt^n..m.

Lett⁰₀ be a ΣQ-vector, ˆt⁰₁ a (Σ₁\Σ)Q-vector, t^1..m₁ ^T1 a sequence of Σ₁Q- vectors, ˆt⁰₁₂ a (Σ12\Σ)Q-vector, and t^1..m₁₂ ^T12 a sequence of Σ12Q-vectors. By Theorem 3, the condition ‘T12 is a deductive conservative extension of T1’ can be represented by means of the following closed quantified Boolean formula

∀

∃

∃

where, for a TBoxT andN ≥m_T,

φT(t^0..N) =

N

^

j=0

θT(t^j) ∧

mT

^

i=1

%P_i,i(t^0..N{∃Pi,∃P_i⁻}),

θ_T(t) = ^

D₁vD2∈ T

t(D₁)→t(D₂) ,

%P,i(p^0..N) = pⁱ(∃P)→_^N

j=0p^j(∃P⁻)

∧ pⁱ(∃P⁻)→_^N

j=0p^j(∃P)

∧ ¬pⁱ(∃P)∧ ¬pⁱ(∃P⁻) → ^^N

j=0,j6=i ¬p^j(∃P)∧ ¬p^j(∃P⁻) .

Theorem 4. For each assignment a, we have a(φ_T(t^0..N)) = T iff the set {t^a(t⁰), . . . ,t^a(t^N)}of sig(T)Q-types is preciselyT-realisable in a modelI where P_i^I6=∅iffa(tⁱ(∃P_i)) =Tora(tⁱ(∃P_i⁻)) =T, for1≤i≤m_T. In particular,T₁₂ is a deductive conservative extension ofT1 w.r.t.Σ iff QBF (1)is satisfiable.

There are different ways of transforming (1) into a prenex CNF, which is a standard input to QBF solvers (seehttp://dcs.bbk.ac.uk/~roman/qbf for some options). One of the versions we used in our experiments is of the form

∀

∀

∃

∃

∃

∃

h

φ⁰_T1(t⁰₀·ˆt⁰₁,t^1..m₁ ^T1,u1. . .umT1,w^0..mT1, p) ∧ φ⁰⁰_T12(t⁰₀·ˆt⁰₁₂,t^1..m₁₂ ^T12, p)i ,

whereu1, . . . ,um_T1,w^0..mT1 andpareKauxiliary variables,K= (m_T1+1)C_T1+ 3m_T1+ 1 andC_T is the number of axioms in T. In total the prenex QBF has (m_T1+1)W_T1universal and (m_T12+1)W_T12−W0+Kexistential variables, where W_T andW0 are the numbers of basic concepts inT andΣ, respectively. CNFs φ⁰_T(t^0..N,u1. . .umT,w^0..N, p) andφ⁰⁰_T(t^0..N, p) contain (N+ 1)B_T + 1 + (2N+ 7)mT and (N+ 1)(CT +B_T⁰ ) + 2(N+ 1)mT clauses, whereBT andB_T⁰ are the numbers of basic concepts in the left- and right-hand sides inT, respectively.

The order of the variables in the prefix has a strong impact on the solvers’

performance (as is well-known in the QBF community), and usually one can fine- tune it depending on the solver. Another important parameter, which has not been studied comprehensively yet by the QBF community, is the structure of the prefix. For example, some of the existential quantifiers can be moved right after the universal ones they depend on, which gives a prefix of the form∀∃. . .∀∃. The impact of this transformation is not completely clear. However, our experiments show—especially for the more complex query conservativity—that the structure of the prefix may become crucial for a solver to succeed.

3.1 Experimentation

We experimented with several variants of the above translation. As our bench- marks, we considered three series of ‘3D’ instances of the form (Σ,T1,T12), with Σ containing 1–10 roles and 5–52 basic concepts,T1 containing 8–25 roles, 47–

122 basic concepts, 59–154 axioms, and T12 9–30 roles, 49–147 basic concepts and 74–198 axioms. In all instances of the first series, theNN-series,T12is not a deductive conservative extension ofT1w.r.t.Σ; in theYN-series, it is a deductive but not a query conservative extension; and in theYY-series,T12is a query con- servative extension ofT1 w.r.t.Σ. The reader can find the benchmarks (as both L^ATEX and .qdimacs files, as well as a .qdimacs translator for L^ATEX files) athttp:

//dcs.bbk.ac.uk/~roman/qbf. It is to be noted that our ontologies arenotran- domly generated. On the contrary, we use ‘typical’DL-Liteontologies available on the Web: extensions of DL-Lite_bool fragments of the standard ‘department ontology’ (as in Example 1) as well asDL-Lite_bool representations of the ER dia- grams used in the QuOnto system (http://www.dis.uniroma1.it/~quonto/).

The number of clauses in the prenex QBFs ranges over the interval 2,000–18,300;

Quaffle2clsQsKizzoT12size

YYtime

0 20 40 60 80 100 120 140 160

50 60 70 80 90 100 110

axioms

12 17 21 30 |Σ|

YNtime

0 20 40 60 80 100 120 140 160

60 80 100 120 140 160 180

axioms

14 18 33 43 |Σ|

NNtime

0 20 40 60 80 100 120 140 160

80 100 120 140 160 180 200

axioms

15 30 40 50 |Σ|

Fig. 1.Run-times for deductive conservativity.

the number of universal variables ranges over 340–3,200 and the number of ex- istential variables over 900–8,600.

We checked conservativity for our benchmarks with the help of three state- of-the-art QBF solvers: sKizzo [7], 2clsQ [8] and Quaffle [9, 10]. The tests were conducted on a P4 3GHz machine with 2GB memory (in fact, ≤300MB was required). The detailed results of the experiments are available athttp://dcs.

bbk.ac.uk/~roman/qbf. Here we only give a very brief summary; see Fig. 1.

The only solver to cope with all 828 instances was Quaffle. In the most complex cases, Quaffle needed 32.8 s to solve an NN instance with |Σ| = 52,

|T12|= 198 and 18,277 clauses, 11,809 variables in the translation (3,172 univer- sal and 8,585 existential); and 224.4 s to solve an YN instance with|Σ| = 45,

|T12|= 191 and 17,424 clauses, 11,374 variables in the translation (∀³⁰⁰⁰∃⁸³²⁹).

The big difference in the run-time for these two instances may be explained by the fact that the former only required a counterexample, while the latter needed an analysis of the whole search space. The overall performance of Quaffle was by far the best in the case of deductive conservativity. On the YY series, sKizzo per- formed much better than both Quaffle and 2clsQ (except two instances), while on the NN and YN series sKizzo was rather poor.

4 Query conservativity

The worst-case complexity of query conservativity is the same as the complexity of deductive conservativity: both are Π₂^p-complete. However, the query conser-

vativity criterion from [6] looks much more involved: unlike Theorem 2 dealing with realisability of individualtypes, now we have to deal withsets of types.

Theorem 5 ([6]).T₁₂is a query conservative extension ofT₁w.r.t.Σiff every preciselyT1-realisable set of ΣQ-types is preciselyT12-realisable.

To make this criterion more efficient, we observe first that a setΞofsig(T)Q- types is preciselyT-realisable iff every type inΞ has aT-witness set within Ξ.

So the following conditions are equivalent:

– T12is a query conservative extension ofT1 w.r.t.Σ;

– for everyT₁-witness set Θ^T_t¹ for a ΣQ-type t, the set Θ_t^T1 Σ is precisely T12-realisable, whereΘ^T_t¹Σ is the set of restrictions of types inΘ^T_t¹ to Σ.

Intuitively, this result means that we do not have to consider arbitrary sets of Σ1Q-types, but only those of size ≤ mT₁ + 1 that are ‘generated’ by a ΣQ- type t and ordered in such a way that a certain typeti in the ordering ‘takes care of P_i.’ Now we extend the notion of a T-witness set as follows. For aT1- witness setΘ^T_t¹ =t₀,t₁, . . . ,t_mT

1 andM =m_T12−m₀, call a sequenceΘ^T_t^1T12 = ˆt0,ˆt1, . . . ,ˆtmT1,s1, . . . ,sM ofΣ12Q-types aT12-witness set for Θ_t^T1 if

(a2) for each 1≤i≤m_T1,tiΣ⊆ˆti,

(a⁰₂) for each 1≤j≤M, there is 1≤k≤mT₁ withtkΣ⊆sj, (b2) each type in ˆt0,ˆt1, . . . ,ˆtmT1,s1, . . . ,sM isT12-realisable,

(c₂) ∃P_i ∈ˆt_j, for some 1 ≤ j ≤ m_T1, or ∃P_i ∈ s_k, for some 1 ≤k ≤ M, iff

∃Pi ∈si or ∃P_i⁻∈si, for each role namePi inΣ12\Σ, 1≤i≤M.

Theorem 6. T12 is a query conservative extension of T1 w.r.t.Σ iff, for every T1-witness setΘ_t^T1 for some ΣQ-type t, there is a T12-witness set for Θ^T_t¹.

In the criterion of Theorem 3, we had to take aΣQ-typet, (i) extendtto a Σ1Q-type, (ii) check whether there are ‘witnesses’ for all the roles in that type and the types providing those witnesses, and if this is the case, we finally had to repeat steps (i) and (ii) again forΣ12in place ofΣ1. The criterion of Theorem 6 is much more complex not only because now we have to start with a set of (mT₁+ 1)Σ1Q-types rather than a single type. More importantly, now the T12

witnesses we choose for these types are not arbitrary but must have the same Σ-restrictions as the original Σ₁Q-types. This last condition makes the QBF translation much more complex (see below) and, consequently, computationally more costly.

LetM =m_T12−m0,t^0..m₀ ^T1 beΣQ-vectors,ˆt^0..m₁ ^T1 (Σ1\Σ)Q-vectors,tˆ^0..m₁₂ ^T1, s^1..M₁₂ be (Σ12\Σ)Q-vectors. By Theorem 6, the condition ‘T12 is a query con- servative extension ofT1’ can be expressed by the following closed QBF

∀

∃

∃

∃

whereβT₁₂(t^0..m₀ ^T1,ˆt^0..m₁₂ ^T1,s^1..M₁₂ ) is the formula

m_T1

^

j=0

θ_T12(t^j₀·tˆ^j₁₂) ∧

M

^

j=1 m_T1

_

k=0

θ_T12(t^k₀·s^j₁₂)

∧

M

^

i=1

%P_{m0 +i},i((s^1..M₁₂ ){∃Pi,∃P_i⁻},(ˆt^0..m₁₂ ^T1){∃Pi,∃P_i⁻}), andφ_T,θ_T and%_P,i are defined as before (here we assume that all concepts∃R forΣ-roles precede those forΣ₁₂\Σ-roles).

Theorem 7. T12 is a query conservative extension of T1 w.r.t. Σ iff (2) is satisfiable.

It can be checked that (2) is equivalent to the prenex QBF

∀

∃

∃

∃

∃

∀

∃

∀

∃

∃

h

φ⁰_T1((t⁰₀·ˆt1)^0..mT1,u1. . .umT1,w^0..mT1, p)∧

β_T⁰⁰₁₂(t^0..m₀ ^T1,ˆt^0..m₁₂ ^T1,s^1..M₁₂ ,q^0..mT1, p)i ,

where q^j = (q^j₁, . . . , q^j_M), for 0 ≤j ≤ mT₁, φT is as before and β_T⁰⁰ is a CNF equivalent to (p→βT). The latter CNF contains (M+ 1)(N+ 1)(CT +B_T⁰ ) + 2M(M+N+ 1) +M clauses, whereN =m_T1, which is quadratic inm_T12, the number of roles inΣ₁₂(unlikeφ⁰⁰_T

12, which is only linear inm_T12).

4.1 Experimentation

We checked query conservativity of the same three series (NN, YN, YY) of onto- logies as in Section 3.1, where only in the YY seriesT₁₂was a query conservative extension of T1 w.r.t.Σ. Thus, the 828 instances (Σ,T1,T12) are precisely the same as before. However, their QBF translations are quite different in the query conservativity case: now they have 5,121–153,482 clauses with 344–3,172 univer- sal and 928–11,125 existential variables. Unfortunately, not all of the instances have been solved by the three solvers. For the detailed results of the tests the reader is again referred to http://dcs.bbk.ac.uk/~roman/qbf, while here we confine ourselves to a brief summary; see Fig. 2.

Quaffle, the deductive conservativity ‘champion,’ could not solve a single query conservativity instance in 300 s. 2clsQ showed a reasonable and, more importantly, robust performance in the NN and YN cases. The most complex YN instance it solved in 420 s had the following parameters:|Σ|= 45,|T12|= 176, 100,334 clauses in the QBF translation with 2,576 universal and 7,674 existential variables. On the other hand, 2clsQ could not solve any YY instances with timeout 300 s. sKizzo’s performance was poor on both NN and YN instances, where only very few instances were solved. However, quite unexpectedly, for some variant of the translation, sKizzo managed to solve about 80% of YY instances (in 4200 s), where the two other solvers completely failed.

Quaffle2clsQsKizzoT12size

YYtime

0 500 1000 1500 2000 2500 3000

50 60 70 80 90 100 110

axioms

12 17 21 30 |Σ|

YNtime

0 100 200 300 400 500 600

60 80 100 120 140 160 180

axioms

14 18 33 43 |Σ|

NNtime

0 100 200 300 400 500 600

80 100 120 140 160 180 200

axioms

15 30 40 50 |Σ|

Fig. 2.Run-times for query conservativity.

5 Conclusion

The empirical results presented above indicate that it is indeed possible to auto- matise conservativity checking for DL-Lite ontologies using off-the-shelf QBF solvers. The main application area of the DL-Lite family of logics is concep- tual data modelling and data integration, where typicalDL-Lite ontologies do not contain more than a few hundred axioms. We have seen that modern QBF solvers can easily check deductive conservativity for ontologies of this size, even without any strategies, variable orderings, or other techniques developed spe- cially for this particular case. It turned out, however, that query conservativity is usually more time demanding and would clearly benefit from some special QBF techniques for quantifier ordering, variable elimination or BDD ordering.

(In fact, the QBF instances generated from our ontologies can form new and unusual benchmarks for QBF solvers.) For example, we plan to experiment with the AQME solver, which can inductively learn its solver selection strategy [14].

Other interesting directions for future research include the following:

– It was shown in [6] that for sub-Boolean variants ofDL-Litedeciding conser- vativity becomes ‘only’coNP-complete. It would be of interest to consider algorithms for this simpler case as well as other notions of conservativity.

– One can also consider sound but incomplete algorithms such as the locality- based approach of [3]. Unfortunately, very few of the test cases considered

in this paper are in the scope of the locality-based approach, yet we believe that new approximations can be developed.

– We plan to extend our results to the slightly more general case whereT₁and T12 are incomparable. Checking whether they have the same consequences over a given signature can then be regarded as a logic-based generalisation of the standarddiffoperator for different versions of texts.

– In applications, it is important not only to know thatT12is not a conservative extension ofT1, but also to have a corresponding counterexample. It remains to be investigated how such counterexamples can be generated using the algorithms presented in this paper.

Acknowledgements. We are grateful to Marco Benedetti for his help.

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